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π in Other Universes

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Re: π in Other Universes

#71

Post by quickthrower2 »

codeflo wrote: post #69
nlitened wrote: post #63

Earlier quoted context omitted.

I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for exampl…

You seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline. exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ... (BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't…

I had never spotted that before, each term of the series is the integral of the previous. That is pleasing!

The Pi thing feels now less of a coincidence than the fact that exp is a power. That probably falls out of expanding the polynomials but it so ingrained as taken for granted that it is wonderous when you think about it.

Re: π in Other Universes

#72

Post by BlueTemplar »

codeflo wrote: post #36

* pi = 3.14159… appears in analysis and by extension statistics, independent of geometry. So aliens in these other universes would know this value, they’d just have a different constant for circles. Since they wouldn’t use Greek letters anyway, we’d have to translate, and it would be a bit silly to equate their 3.757… with “pi” instead of their 3.14159… * Personal aside: Of course, whether 3.14… (pi), 6.28… (2pi) or…

Yes, and they actually keep using 2pi over and over in their examples.

Re: π in Other Universes

#73

Post by BlueTemplar »

wcoenen wrote: post #60

I noticed that all the "circles" for alternative metrics are aligned with the coordinate system. For example, the one for the Manhattan distance has its corners on the coordinate axes. What if we added an additional condition that a distance metric should not change when the orientation of the coordinate system is changed? Could we still have different values for the pi constant then?

Is that true for the hexagon, or just very close ?

Re: π in Other Universes

#74

Post by lloeki »

There's this fun space made of p-adic numbers upon which you can define a simple distance, and then circles have mind bending properties like the diameter (max edge to edge distance) and radius (distance from edge to center) being equal to each other.

Quirky stuff happens to disc area and perimeter as well, and open discs are also closed. The equivalent of Pi there is nuts.

Sadly I can't recall the details (it was a 2000-ish exercise on my maths course).

https://en.wikipedia.org/wiki/P-adic_number#Topological_prop...

Re: π in Other Universes

#75

Post by lmpdev »

tetha wrote: post #51
IanCal wrote: post #2

> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable This is a really, really n…

This is also a part of why I am somewhat fascinated by the idea and the state of Lean4 and mathlib in Lean4. People put more and more formally verified proofs into mathlib, which in turn makes formally proving further theorems in mathlib easier. If you start with nothing (like in the numbers game), simple proofs are a lot of ... just effort, because you have specify a lot of rewrites and overall work. In mathlib, how…

Former computational mathematics major

You absolutely can contribute meaningfully

The maths world is incomprehensibly broad and deep, even if you just take the Erdos approach and go for interesting but shallow problems

Re: π in Other Universes

#76

Post by nlitened »

codeflo wrote: post #69
nlitened wrote: post #63

Earlier quoted context omitted.

I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for exampl…

You seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline. exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ... (BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't…

Hey, you're right! So Pi is special :)

Re: π in Other Universes

#77

Post by nlitened »

quickthrower2 wrote: post #70
nlitened wrote: post #63

Earlier quoted context omitted.

I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for exampl…

I guess radians are “magic” in that sin and cos can be defined by infinite series that look nice (and feel canonical). You have to manipulate those series to get 360 or even just revolutions.

As codeflo showed in a sibling comment, actually I am wrong, and Pi's special magic also becomes from Taylor's series expansion. So turns out that Pi is the real magic number rather than just our arbitrary choice!

Re: π in Other Universes

#78

Post by nlitened »

messe wrote: post #67
nlitened wrote: post #62

Earlier quoted context omitted.

As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*

The only function that satisfies those two equalities is e^ix which has period 2π.

Hey, you're actually right! My bad

Re: π in Other Universes

#79

Post by mannykannot »

dclowd9901 wrote: post #5

I must be missing something. In the example of using a sailboat with constant wind and distance, wouldn’t sailing against the wind (let’s call it any constant oppositional force), cause us to get a circle, just shifted from the origin? Not an ellipsis?

There are a number of complications. Two of them are that firstly, when when the wind direction is within about 45 degrees against the direction you want to go, you have to tack, and secondly, a reasonably efficient sailboat is fastest when it is on a reach, with the wind coming from the side.

https://physics.stackexchange.com/questions/186515/why-is-a-...

Re: π in Other Universes

#80

Post by jojobas »

nlitened wrote: post #62
moring wrote: post #46

Earlier quoted context omitted.

Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results. note 1: exp(x) can alternatively be defined by the exponential series, but that series…

As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*

Consider simultaneous functional equations:

f(x) = dg(x)/dx

g(x) = df(x)/dx

Its only linearly independent solutions are sin(x+c) and cos(x+c) with, x being in radians, periodic by 2pi.

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